Recursive Bayesian estimators such as Kalman filters (KFs) are used to estimate an unknown probability density function (PDF) recursively. For example, extended Kalman filters are used to perform state estimation of non-linear systems. The KF-based estimators provide system state estimates in the form of the conditional moments on the basis of system model and measurements.
The filters cyclically perform two steps: the prediction step and the filtering step. If the estimation performance of the KF-based estimators is not sufficient, other estimators such as a point-mass filter (PMF) or a Rao-Blackwellized point-mass filter (RBPMF) can be used. These non-linear filters, which are based on the numerical solution to the Bayesian recursive relations (BRRs), provide state estimates in the form of the probability density functions (PDFs).
In the prediction step, the model of the non-linear system is used to predict (or to estimate) a future value of one or more states. In the filtering step, measurement data is used to correct the predictive estimate, and generates a filtered estimate of the states. The filtered estimate of the states is typically more accurate than the predictive estimate of the states because of the use of the measurement data.
The numerical solution based recursive Bayesian estimator is based on an approximation of a discrete probability density function using a point-mass density. Discrete PDFs, i.e., the point-mass densities, are formed over a grid of points, or an array of grid points. A probability is assigned to each grid point. While discrete PDFs are more readily analyzed with modern computers, they can give rise to approximation error.
The properties of the grid determine the accuracy of the numerical solution of the recursive Bayesian estimator. The properties include number of points, distance between the points, space (e.g. distance, area and volume), shape, and location of the grid.
Generally, the more grid points used, the more accurate the estimate provided by the Bayesian estimator. However, even with modern computation systems, the number of points that can be used is limited. Therefore, there is a need to enhance the accuracy of the Bayesian estimate other than by increasing the number of grid points.